# Thursday, January 13

:::{.remark}
Last time: fields are simple rings.
:::

:::{.exercise title="?"}
Let $k\in \Field$ and
show that

- If $f\in R\da k[x]$ is irreducible then $\gens{f} \in \spec R$.
- $\gens{xy} \in \Id(k[x,y])$ is not prime and not maximal.
- There exist nonzero non-prime maximal ideals.
- Show that $I\in \spec R \iff A/I$ is an integral domain.
- Show that $I\in \mspec R \iff A/I\in \Field$.

:::

:::{.exercise title="?"}
Prove that if $f: R\to S$ is a ring morphism then there is a well-defined map
\[
f^*: \spec S &\to \spec R \\
\mfp &\mapsto f\inv(\mfp)
,\]
i.e. if $\mfp$ is prime in $S$ then $f\inv(\mfp)$ is prime in $R$.
Show that this doesn't generally hold with $\spec$ replaced by $\mspec$.
:::

:::{.exercise title="?"}
Show that defining $V(I) \da \ts{\mfp\in \spec R \st \mfp\contains I}$ as closed sets defines a topology on $\spec R$.
Show that $\spec R$ is Hausdorff iff it's discrete, and $V$ is functorial.
:::

:::{.exercise title="?"}
Describe

- $\spec k$ for $k\in \Field$
- $\spec \ZZ$, show it is not Hausdorff, and $\gens{0}$ is a generic point, and the subspace topology on its closed points is cofinite.
- $\spec R$ for $R$ a local ring.
- $\spec R$ for $R$ a DVR
- $\spec k[x,y]$
- $\spec R$ for $R = \ZZ[i]$.
- $\spec \OO_K$, the ring of integers of a number field $K$.
- $\spec R$ for $R$ a Dedekind domain

:::

:::{.exercise title="?"}
Show that every $I\in \Id(R)$ is contained in some $\mfm \in \mspec R$.
:::

:::{.exercise title="?"}
Show that the following rings are local:

- For $p\in \ZZ$ prime, $R\da \ZZ\localize{S}$ for $S\da\ts{\ell \in \ZZ \text{ prime} \st \ell \neq p}$.
- $k\in \Field$
- $k\powerseries{x}$ with $\mfm = \gens{x}$
- $k\powerseries{x, y}$.
  What is the maximal ideal?

:::

:::{.exercise title="?"}
For $(R, \mfm)$ a local ring, show that $\mfm = R\sm(R\units)$. 
:::

:::{.remark}
Recall that

- $\sum I_j$ is the smallest ideal containing all of the $I_j$.
- $\Intersect I_j$ is again an ideal
- $IJ \da \gens{xy \st x\in I, y\in J}$ is an ideal
- $IJ \subseteq I \intersect J$.
- $\nilrad{R}$ is the set of nilpotent elements.

:::

:::{.theorem title="?"}
Show that $\nilrad{R}$ is the intersection of all prime ideals.
:::

:::{.slogan}
Regarding elements $f\in R$ as functions on $\spec R$, $f$ nilpotent is like being zero at every point of $\spec R$.
:::

:::{.exercise title="?"}
Show that $x\in \jacobsonrad{R} \iff 1-xy\in R\units$ for all $y\in R$.
:::